Stable/unstable manifolds transformation

FIGURE 9 Stroboscopic phase portraits for the points on figure 8 labelled (a)-(e). (a) Below the third root of unity point (labelled F in figure 8) the phase portrait is structurally a period three phase locked torus (b) above point F, the period I fixed point in the centre is now stable and the phase locked torus has disappeared (c)-(e) before, during, and after a period 3 homoclinic bifurcation to the right of point F oil cut, = 3.97 for each, and A/Ao = 5.90, 5.93 and 5.95 for (c)-(e) respectively. The period 3 phase locked torus is transformed to a free torus as the stable manifold of each saddle crosses the unstable manifold of an adjacent saddle. 

Wiggins et al. [22] pointed out that one can always locally transform a Hamiltonian to the form of Eq. (1.38) if there exists a certain type of saddle point. Examination of the associated Hamilton s equations of motion shows that q = Pn = 0 is a fixed point that defines an invariant manifold of dimension 2n — 2. This manifold intersects with the energy surface, creating a (2m — 3)-dimensional invariant manifold. The latter invariant manifold of dimension 2m — 3 is an excellent example of an NHIM. More interesting, in this case the stable and unstable manifolds of the NHIM, denoted by W and W ,... 

It is well known (see, e.g.. Ref. 13) that the normal form transformations do not converge in the sense that normalization to all orders generally does not yield a meaningful result. However, this is of no consequence for our purposes. We view the technique more as the input to a numerical method for realizing the NHIM, its stable and unstable manifolds, and the TS. In this sense the limitations of machine precision make normalization beyond a certain finite order meaningless. This is a local result valid in the neighborhood of the equilibrium point of center center saddle type. However, once the phase-space structure is established locally, it can be numerically continued outside of the local region. 

This is also the Hamiltonian of the activated complex. We will encounter it in Eq. (23) with the customary symbol H. Regardless of its stability properties or the size of the nonlinearity, Eq. (12) is always an invariant manifold. However, we are interested in the case when it is of the saddle type with stable and unstable manifolds. If the physical Hamiltonian is of the form of Eq. (1), then a preliminary, local transformation is not required. The manifold (12) is invariant regardless of the size of the nonlinearity. Moreover, it is also of saddle type with respect to stability in the transverse directions. This can be seen by examining Eq. (1). On qn = Pn = 0 the transverse directions, (i.e., q and p ), are still of saddle type (more precisely, they grow and decay exponentially). 

First, transform the variables to read (A , y) = (a — A o(y),y), and let (A , y) denote x, y) for simplicity. For the normal directions of the manifold Mq, the eigenvectors of the matrix A = 0,y) offer the linear approximation to the stable and unstable manifolds of Mq the coordinates beyond the linear approximation are obtained by following these eigenvectors in the backward and forward directions in time, respectively. Let a and b denote the coordinates thus obtained along the unstable and stable manifolds, respectively. Then, the stable manifold Wq is given by a = 0, and the unstable manifold Wq is given by b = 0 and x = (a,b). A schematic picture of these manifolds is shown in Fig. 5. 

Since the manifold Mq is a NHIM, it changes continuously, under a small perturbation, into a new NHIM M - Moreover, the separatrix Wq changes, continuously and locally near M, into the stable manifold and the unstable one W" of the NHIM M. Note, however, that, in general, and W no longer coincide with each other to form a single manifold globally. Then, the Lie transformation method brings the total Hamiltonian H x.I, 0) into the Fenichel normal form locally near the manifold M. ... 

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